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Example 6: Find a polynomial of degree 3 such that 1, and – 2 are zeros, 1 has a multiplicity of 2 and
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The Fundamental Theorem of
AlgebraIntro - Chapter
4.6
• If every zero is counted as many times as its ______________ then, a polynomial of degree n has __________ n complex zeros.
The Fundamental Theorem of Algebra
Every non-constant polynomial has a ______ in the complex number system.
ZERO
AT MOST• Every polynomial of degree n > 0 has __________ n different _______ in the complex number system.
ZEROS
MULTIPLICITYEXACTLY
Example 6: Find a polynomial of degree 3 such that 1, and – 2 are zeros, 1 has a multiplicity of 2 and 2 32f
21 2f x a x x
f x
232 2 1 2 2a 32 4
8a
a 28 1 2f x x x
Conjugate Zero TheoremLet be a polynomial with ______ coefficients. If the complex number z is a zero of ,then its _____________, ____, is also a zero of .Example 7: Find a polynomial with real coefficients whose zeros include the numbers 2 and 3+ i
CONJUGATE
REAL
z
2 3 3a x x i x i
f x f x f x
22 3 3 9 1x x i x i x
22 3 3 10x x x ix x ix
22 6 10x x x 3 28 22 20x x x
1let a
Example 8: Factor
4 1 5 4 2 8
4 4 0 8
1 1 0 2 0
3 24 2x x x
All possible zeros: 1, 2, 4, 8 1 1 1 0 2
1 2 2
1 2 2 0
8245 234 xxxxxf
24 1 2 2f x x x x x
22 2 4 2 2 4 2 2 12 2 2
ix i
4 1 1 1f x x x x i x i
Real plane:
Complex plane:
